Predicting two-dimensional turbulence

Prediction is a fundamental objective of science. It is more difficult for chaotic and complex systems like turbulence. Here we use information theory to quantify spatial prediction using experimental data from a turbulent soap film. At high Reynolds number, $\text{Re}$, where a cascade exists, turbulence becomes easier to predict as the inertial range broadens. The development of a cascade at low $\text{Re}$ is also detected.

Figure 1

Plot of the fundamental quantities $h (◯),E (\square )$, and $C \left(▵\right)$ for the simple example given here. Although $h$ and $E$ are continuous functions of $P,C$ is not.

Figure 2

Left: Experimental setup showing the reservoirs $(TR,BR)$, pump $\left(P\right)$, valve $\left(V\right)$, comb $\left(C\right)$, blades $(LB,RB)$, LDV, and weight $\left(W\right)$. Middle: Fluctuations in film thickness from turbulent velocity fluctuations with smooth walls and a comb. Right: Thickness fluctuations with smooth and rough walls.

Figure 3

Representative one-dimensional energy spectra in a log-log plot of $\mathcal{E}\left(k\right)$ vs $k$. The enstrophy cascade $\left(▵\right)$ has a slope close to $-3$ while the energy cascade $(\square )$ has a slope close to $-5/3$. The flat curve $(◯)$ has no cascade.

Figure 4

The statistical complexity $C (\square )$, excess entropy $E (◯)$, and entropy density $h \left(▵\right)$ as functions of $\text{Re}$ for binarized $\left(A=2\right)$ data (see Appendix pp1 for details on binning). We plot $h$ on a different scale for better visibility. The maximum value of $h$ here is ${log}_{2}2=1$, which the no-cascade data for $\text{Re}<100$ approach very closely. Here $L=10$ and we used our matlab program with the ${\chi }^2$ test to calculate $C$ (see Appendix pp4 for details). The lines are not fits to the data but are meant to suggest the behavior of $C$ and $E$ as functions of $\text{Re}$. For the cascade region, $C$ and $h$ are decreasing functions of $\text{Re}$ while $E$ increases. The vertical line separates the data according to whether there is a cascade or not.

Figure 5

The velocity autocorrelation function $c\left(r\right)$ plotted vs $r$ for several values of $\text{Re}$. For small $\text{Re},c\left(r\right)$ quickly decays to zero, indicating little correlation in the velocity $u$. For larger $\text{Re}$, where Fig. 3 indicates spatial structure, there is a wider range of correlated scales. The $\text{Re}=300$ curve has a longer correlation length $L$ than the higher $\text{Re}=6000$ curve presumably because this lower $\text{Re}$ curve corresponds to an inverse energy cascade. The inverse energy cascade is supposed to involve larger length scales than the enstrophy cascade [23, 28]. Here $L$ is defined as the distance at which $c\left(r\right)$ decays to $1/e$.

Figure 6

The predictive efficiency $E/C$ plotted vs $\text{Re}$ using the same data as in Fig. 4 as well as a quaternary partition $A=4$ with partition walls placed symmetrically with respect to the mean (see Appendix pp1 for details on binning). We used $L=10$ for both partitions (see Appendix pp4). Here we find that $E/C$ is increasing only after a cascade develops.

Figure 7

An example of three conditional probability distribution functions used to determine the causal states. The data used here is binarized turbulence data with $L=5$ (giving a total of 32 possible states) and $\text{Re}=3300 (◯=00001,+=00011,▵=00010)$. The horizontal axis features all the possible future states while the vertical axis is the conditional probability that given a certain past state, any of those possible future states will occur. Here the distribution for states $◯$ and $\square$ appear similar while that for state $▵$ is quite different.